线性衰减而不是指数衰减：一个数学模型和基本机制
Linear rather than exponential decay: a mathematical model and underlying mechanisms
摘要：
一些群体，如红细胞（RBC），表现出更接近线性而不是指数的群体下降模式，这已经证明用单个简单的数学模型来描述是出乎意料的挑战。在这里，我们表明，一个亚指数模型的人口灭绝可以很好地近似RBC灭绝的实验曲线，和亚指数衰减的一个可能的机制是人口的异质性相对于死亡率的个人。我们进一步证明了，如果存在异质性的死亡率，使他们的初始分布是伽玛分布的人口下降的亚指数模型可以得到的频率依赖的人口灭绝模型的框架内。值得注意的是，可能导致种群线性衰减模式的特定生物机制可能因特定生物系统而异;然而，最终它们必须收敛于种群内不同的个体死亡率，因为统一的死亡率将导致种群指数下降。因此，所提出的模型并不旨在描述RBC生物学的复杂动力学，而是可以提供一种从现象学上描述种群大小线性衰减的方法。我们简要讨论了这种模型的潜在效用，用于描述可能导致RBC耗竭的药物的作用，并得出结论，建议这种工具可以提供一个镜头，用于发现其他人群中的线性灭绝模式，这可能是以前被忽视的。
英文摘要：
Some populations, such as red blood cells (RBCs), exhibit a pattern of population decline that is closer to linear rather than exponential, which has proven to be unexpectedly challenging to describe with a single simple mathematical model. Here we show that a sub-exponential model of population extinction can approximate very well the experimental curves of RBC extinction, and that one possible mechanism underlying sub-exponential decay is population heterogeneity with respect to death rates of individuals. We further show that a sub-exponential model of population decline can be derived within the frameworks of frequency-dependent model of population extinction if there exists heterogeneity with respect to mortality rates such that their initial distribution is the Gamma distribution. Notably, specific biological mechanisms that may result in linear pattern of population decay may be different depending on the specific biological system; however, in the end they must converge to individual death rates being different within the population, since uniform death rates would result in exponential population decline. As such, the proposed model is not intended to describe the complex dynamics of RBC biology but instead can provide a way to phenomenologically describe linear decay of population size. We briefly discuss the potential utility of this model for describing effects of drugs that may cause RBC depletion and conclude with a suggestion that this tool can provide a lens for discovering linear extinction patterns in other populations, which may have previously been overlooked.
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